Polynomiography is the art and science of visualization in approximation of zeros of complex polynomials, via fractal and non-fractal images created using the mathematical convergence properties of iteration functions. An individual image is called a polynomiograph, and represents a certain graph of polynomials, not in the conventional sense of plotting a graph (e.g., a parabola for a quadratic polynomial). Polynomiographs are obtained using algorithms that require the manipulation of thousands of pixels on a computer monitor. Depending upon the degree of the underlying polynomial it is possible to obtain beautiful images on a laptop computer in less than the running time of a TV commercial.
Polynomials form a fundamental class of mathematical objects with diverse applications and arise in devising algorithms for such mundane tasks as multiplying two numbers, much faster than in conventional ways. According to the Fundamental Theorem of Algebra, a polynomial of degree n, with complex coefficients, has n zeros (roots) which may or may not be distinct. The task of approximation of zeros of polynomials is a problem that was known to Sumerians in the third millennium B.C. This problem has been one of the most influential problems in the development of several important areas in mathematics.
The word “fractal,” which partially appears in the definition of polynomiography, was coined by the research scientist Benoit Mandelbrot. It refers to sets or geometric objects that are self-similar and independent of scale. This means there is detail on all levels of magnification. No matter how many times one zooms in, one can still discover new details. Some fractal images can be obtained via simple iterative schemes leading to sets known as the Julia set and the Mandelbrot set. The simplicity of these iterative schemes, which may or may not have any significant purpose in mind, has resulted in the creation of numerous web sites in which amateurs and experts exhibit their fractal images.
It would be advantageous to provide a method of creating graphical artwork based on polynomial equations.
It would also be advantageous to use iterative functions to obtain roots of such polynomial equations.
It would still further be advantageous to visualize a polynomial by means of approximating the roots thereof.